and Moho Topography on P Geometric Spreadingthorne/TL.pdfs/ALXY_Pn_BSSA_2011.pdf · Effects of 2D Random Velocity Heterogeneities in the Mantle Lid and Moho Topography on P n Geometric

Effects of 2D Random Velocity Heterogeneities in the Mantle Lid

and Moho Topography on Pn Geometric Spreading

by Megan Avants, Thorne Lay, Xiao-Bi Xie, and Xiaoning Yang

Abstract Pn-wave energy refracts through the uppermost mantle, with the firstseismic wave arrival at distances of ∼200 to ∼1500 km from crustal sources. ThePn phase provides important constraints on source type, location, and magnitude,but its propagation is complicated by frequency-dependent sensitivity to the Earth’ssphericity and lithospheric velocity structure. Converging on an acceptable Pn

geometric spreading correction and specifying its uncertainties, a requirement foraccurately determining frequency-dependent attenuation models for Pn, dependson improved understanding of the behavior of Pn geometric spreading for variousheterogeneous models. We investigate the effects of radial mantle lid velocity gradi-ents, mantle lid random volumetric velocity heterogeneities, and Moho topography onPn geometric spreading using reflectivity and two-dimensional (2D) finite-difference1-Hz wave propagation calculations for elastic Earth models. Mantle lid velocity gra-dients systematically modify the frequency-dependent geometric spreading from thatfound for models with constant velocity but retain the same overall functional form.Pn amplitudes are also sensitive to the presence of modest 2D random lateral velocityheterogeneities within the uppermost mantle, with geometric spreading approaching apower-law behavior as the root mean square strength of heterogeneity increases. 2DMoho topography introduces scatter into the amplitude of Pn, but the overall behaviorremains compatible with that for a laterally homogeneous model. Given the lack ofknowledge of specific small-scale structure for any particular Pn path, the preferredgeometric spreading parameterization is the frequency-dependent model for a constantmantle lid velocity structure unless Pn travel-time branch curvature can constrain theradial gradient in the mantle lid.

Introduction

Distinguishing small earthquakes from low-yield under-ground nuclear explosions (given a typical dearth of teleseis-mic observations) requires that regionally detected seismicphases (Pn, Sn,Pg,Lg) be understood in detail. Several sourcediscrimination techniques have been developed that measurethe relative amount of P=S energy in regional signals (e.g.,Walter et al., 1995; Hartse et al., 1997), with regional tra-vel-time tomography and attenuation models being used tosuppress the effects of wave propagation in the heterogeneouscrust. The contribution of geometric spreading to the regionalseismic signals must also be quantified and accounted for,particularly for use in developing attenuation models.Geometric spreading can be very difficult to distinguish fromanelastic attenuation, especially for Pn and Sn phases, whichare known to have frequency-dependent sensitivity to theEarth’s sphericity, upper-mantle velocity heterogeneity, andlower-crustal structure (Buldyrev andLanin, 1965;Hill, 1973;Sereno andGiven, 1990; Taylor et al., 2002).Morozov (2010)gives a detailed discussion on the definition of geometric

spreading, in which he explains that geometric spreadingcan be defined as amplitude decay due to the expansion ofa single wavefront, or that geometric spreading can be con-sidered in such a way as to include the elastic effects of theEarth’s structure on amplitude decay. The single-wavefrontdefinition requires that any amplitude variations from thissimple model be attributed to scattering attenuation. On theother hand, the fully inclusive definition of geometric spread-ing would consider “scattering attenuation” merely as geo-metric spreading in a heterogeneous Earth. Our modelingefforts assume the latter definition of geometric spreadingand seek to characterize the contribution of scattering toobservedPn amplitudes in a purely elasticmedium.Yang et al.(2007) developed frequency-dependent geometric spread-ing relations for Pn and Sn for simple reference velocitystructures with constant velocities in the uppermost mantlelid (the high-velocity layer underlying the Moho and aboveany upper-mantle low-velocity zone) that differ significantlyfrom commonly used frequency-independent power-law

126

Bulletin of the Seismological Society of America, Vol. 101, No. 1, pp. 126–140, February 2011, doi: 10.1785/0120100113

geometric spreading corrections (e.g., Walter and Tay-lor, 2001).

Pn and Sn are generated by crustal sources and havepaths that refract through the uppermost mantle with somesecondary energy multiply reflecting from the undersideof the Moho discontinuity as long as the radial velocitygradient is above the critical negative gradient (e.g., Yanget al., 2007). The distances at which Pn and Sn are observedas first P-wave and S-wave arrivals ranges from their cross-over with Pg and Lg near 200 km to where P-wave andS-wave energy that has dived down to the vicinity of the tran-sition zone overtakes them near 1500 km. The direct raypaths for Pn and Sn as far out as 1500 km distance bottomin the upper 50 km of the mantle lid for typical seismic ve-locity models, and the multiple underside Moho reflectionsbottom even shallower in the mantle lid. These shallowtrajectories make Pn and Sn phases particularly sensitive toEarth’s sphericity and to details of the velocity structure ofthe lower crust and upper mantle through which they propa-gate nearly horizontally. Theoretical calculations of Pn pro-pagation through a spherical body (Buldyrev and Lanin,1965; Hill, 1973), as well as one-dimensional (1D) reflectiv-ity simulations of Pn propagation (Yang et al., 2007) throughlaterally homogeneous layered models, provide ample doc-umentation of Pn sensitivity to the Earth’s sphericity.

An analysis of variance (ANOVA) study conducted byFisk et al. (2008) shows that Pn amplitude observations havethe highest total path variance of the regional phases (Pn, Sn,Pg, Lg) typically used for regional source discriminants. Thisreflects the pronounced sensitivity of Pn to heterogeneouslithospheric structure and the fact that the Pn phase is notcomprised of a suite of arrivals sampling a wide range ofray parameters that intrinsically average out the path effects,as is the case for the other phases (particularly Pg and Lg).The basic wave propagation characteristics of Sn are similarto those for Pn, but the additional P-to-S conversions in thehigh frequency Sn signals appear to somewhat stabilize rmsamplitude measures relative to those for Pn, so we will focuson Pn in this study.

Geometric spreading corrections are typically definedfor simple reference Earth models, motivated by seismicwaveform modeling procedures; while it is clear that a 1Dseismic model is not precisely correct for any path, our lackof knowledge of the actual 3D structure justifies using asimplified reference model about which there is expectedto be a normally distributed variation of observations. Formany phases this is reasonable; teleseismic body waves withturning points in the relatively homogeneous lower mantleappear to have minor scatter relative to 1D frequency-independent geometric spreading assumptions, while thebehavior of complex multiple arrivals like Lg at regional dis-tances can largely be accounted for by simple frequency-independent power-law amplitude decays by virtue of theirintrinsic strong-wave interference (e.g., Yang, 2002). In thisregard, Pn and Sn are important exceptions; even for a verysimple spherical reference model involving a constant-

velocity crustal layer over a constant-velocity upper-mantlelayer, the computed geometric spreading is frequency-dependent, nonuniform with distance, and nonlinearly sen-sitive to perturbations of the structure (Yang et al., 2007).While observed Pn amplitude decay with distance may bewell characterized by a simple power-law function, separat-ing geometric spreading from attenuation effects that controlthe amplitude decay is very difficult when one cannot justifyassuming a frequency-independent geometric spreadingcorrection. Ignoring the frequency dependence of geometricspreading will inject artificial frequency dependence directlyinto attenuation estimates, undermining the latter’s use forpath corrections or geophysical interpretations.

These considerations motivate using as realistic of a Pn

geometric spreading relation as possible; however, the verysensitivity of Pn to spherical structure, which producescomplexity in the geometric spreading, is also likely to yieldsensitivity to small structural perturbations, with non-self-evident behavior. Full waveform calculations are made hereto evaluate the sensitivity of Pn geometric spreading todeviations from a very simple reference model. We consider1D gradients in the mantle lid velocity structure, 2D lateralvolumetric heterogeneities in the mantle lid structure, and 2Dtopographic heterogeneity of the Moho, with the goals ofassessing what practical choice of Pn geometric spreadingis most reasonable and assessing the attendant uncertainties.We find that modest, geologically reasonable structuralheterogeneity produces strong Pn amplitude variability withdistance, which ensures that considerable errors are asso-ciated with any Pn geometric-spreading parameterization.These errors will be manifested in, for example, attenuationmodels for Pn. Adhering to the seismic modeling paradigm,our pragmatic recommendation is to use the complexfrequency-dependent geometric spreading for a very simplebasic Earth model (Yang et al., 2007) rather than an arbitra-rily parameterized model such as a power-law for whichthere is no known associated 1D reference structure, allowingfor large uncertainties on any derived attenuation model thataccompanies it. The geometric spreading approximation fora laterally homogeneous (1D) Earth model from Yang et al.(2007) has the advantages of being consistent with theoreti-cally predicted Pn propagation and corresponding to a spe-cific parameterized velocity model. However, it does havemultiple parameters, each nonlinearly dependent on changesin the velocity model, which makes applying this approxima-tion slightly more complicated than selecting a single decaycoefficient, as is the case for a power-law approximation.We argue the minor added complexity is necessary, as ourmodels show that Pn does indeed have a nonlinear responseto changes in mantle velocity gradient or small-scale struc-ture. Empirically constraining the average mantle lid velocitygradient by first-arrival travel-time curvature can signifi-cantly improve the choice of reference model, but there willstill be significant uncertainties associated with unresolvedsmall-scale heterogeneities in the mantle lid structure.

Effects of 2D Random Velocity Heterogeneities in the Mantle Lid and Moho Topography 127

Methods

Synthetic calculations in this study employ the 1D reflec-tivity method (Müller, 1977; Randall, 1994), as implementedby Yang et al. (2007), and the 2D P-SV finite-difference codeofXie andLay (1994).We add linear radialmantle lid velocitygradients, randomly varying volumetric mantle lid P-wavevelocity (Vp) heterogeneity, or Moho depth heterogeneityto the homogeneous basic Earthmodel (BEM) fromYang et al.(2007) and Sereno and Given (1990). This simple sphericalreference model contains a 40-km-thick crust and a con-stant-velocity mantle (Fig. 1a). For both the reflectivity andfinite-difference methods, we apply an Earth flattening trans-form (EFT) (Chapman, 1973) to the BEM, which yields 1Dstructures having positive velocity gradients with depth(Fig. 1b). All models are purely elastic (or asymptoticallyapproximated as such for the reflectivity calculations asdescribed by Yang et al., 2007), allowing us to isolate theeffects of structural heterogeneity onPn geometric spreading.Perturbations to the BEM are herein discussed in detail foreach class of heterogeneity that we investigate.

The source model is an isotropic point source located ata depth of 15 km. The source-time function for the reflectiv-ity method is a delta function and is a Gaussian derivativesource-time function with amplitude normalized to 1 forthe finite-difference method. For the 1D velocity models withlinear velocity gradients in the mantle lid, we compute bothhigh-frequency reflectivity and lower-frequency finite-difference synthetics. The reflectivity synthetics span therange 0.75–12 Hz, with spectral estimation following theprocedures described in Yang et al. (2007). The dominantfrequency of our finite-difference synthetics is approxi-mately 0.8–1 Hz. The 2D finite-difference modelingapproach cannot achieve the very high frequencies of the1D wavenumber integration method of Yang et al. (2007),and full exploration of frequency-dependent effects of the2D models is beyond our computational limits. However,comparisons of the methods for simple 1D models indicate

that the basic behavior of Pn geometric spreading is accu-rately reproduced by the finite-difference computations for1-Hz signals. We align the synthetic seismograms withrespect to the mantle P-wave velocity (8:1 km=s) and usea fixed-window width of 9 s. The Pn phase onset for theBEM case is approximately 3 s into the fixed window. Thiswindow effectively isolates the Pn coda from contaminationby the Pg phase following it and allows for arrival timeperturbations that result from heterogeneous path structure.The fixed window is only one of several windowing optionscommon in studies of Pn, so we also calculate amplitudes ofPn with a group velocity window (8:2–7:6 km=s) and with avery short fixed window based on Pn onset that approxi-mately captures the first period of the Pn arrival. These threewindowing methods span the range of measurement proce-dures typically used and allow us to consider the nature ofgeometric spreading for an isolated part of the wavefront ver-sus a coda-dominated signal. Amplitudes are calculated asthe rms amplitude of the windowed Pn energy. Correctionsare applied to the calculated rms amplitude to approximate aspherical wavefront (Af � Ao × �1=sqrt�x��, where Af is theamplitude calculated on a spherical wavefront, Ao is the in-itial amplitude, and x is distance in kilometers) and to trans-form the amplitude (Af) into that of a wave traveling within aspherical Earth (unflattening transform—see equation 3 inYang et al., 2007). The Pn amplitude calculations are madein the distance range from 350 to 1500 km to ensure the Pg

arrival does not bias the rms amplitude calculation, althoughthe fixed window is still contaminated by Pg for the closestdistance.

Pn Sensitivity to Radial Mantle LidVelocity Gradients

The radial P-wave velocity structure of the uppermostmantle is seldom well-constrained by observations. The mostdirect constraint is usually provided by curvature of the Pn

travel-time branch as a function of distance (e.g., Morozova

(a) (b)

Figure 1. (a) Crustal and mantle seismic velocity and density values, layer thickness, and conceptual Pn path for the homogeneousspherical basic Earth model (BEM) used by Yang et al. (2007). (b) The associated 1D flat-Earth model, with slight density and seismicvelocity gradients resulting from Earth flattening transform (EFT).

128 M. Avants, T. Lay, X.-B. Xie, and X. Yang

et al., 1999; Myers et al., 2010) if reliable measurements areavailable over a wide enough distance range. Even whenthere is apparent travel-time branch curvature, it is difficultto distinguish between a smooth gradient with depth versusa layered structure with a few or many small velocityincreases. Receiver function methods and surface wave dis-persion inversions sometimes indicate mantle lid structure,but resolution tends to be poor, especially for P velocity,and layered structures are commonly assumed so that anyradial gradients are only approximated. When lacking directconstraint, reference velocity structures often assume con-stant or near-constant velocity in the mantle lid by default,so the BEM model of Yang et al. (2007) and Sereno andGiven (1990) is representative of many choices. However,there are regions in the Earth where the mantle lid-velocitydoes appear to increase with depth, with velocity gradients ofup to 0:004 s�1 according to some studies (e.g., Zhao andXie, 1993; Morozova et al., 1999), and areas of North Africaand Eurasia, for example, which evidence suggests have highlateral variations in mantle gradient (Myers et al., 2010). Thisraises the question of how useful the geometric spreadingrelations for the BEM are in such regions.

The effect of velocity gradient on Pn geometric spread-ing is illustrated in Figure 2. Using the procedure describedin Yang et al. (2007), we computed synthetic Pn seis-mograms for the frequency band of 0.75–12 Hz, using thereflectivity method for models with positive velocity gradi-ents in the mantle lid. The EFT further increases these gra-dients, causing Pn ray paths to turn even more shallowly inthe mantle lid than those for the BEM model. The amplitude–distance behavior is shown for multiple narrow frequencybands for the BEM (red curves) and for models with mantlevelocity gradients of 0:002 s�1 (blue curves) and 0:004 s�1

(black curves). The BEM amplitude curves correspond

directly to those for which the geometric spreading formulasof Yang et al. (2007) were derived. The systematic variationof Pn-amplitude frequency dependence, with increasingamplitudes and a decrease in the distance at which the mini-ma of the amplitude curves occur, is apparent. The overallshapes of amplitude curves for the different models aresimilar, with it being apparent that the effect of positive gra-dient is essentially to shift the curves toward higher frequen-cies relative to the BEM case. The 0:002 s�1 gradient modelproduces 0.75-Hz amplitudes that behave very similarly tothe 9-Hz amplitudes for the BEM, and the 0:004 s�1 gradientmodel produces 0.75-Hz amplitudes very similar to the12-Hz amplitudes for the BEM. This is problematic in thatfailure to constrain the lid-velocity gradient a priori canclearly lead to erroneous frequency dependence of anyderived Pn attenuation model in basically the same way thatusing a frequency-independent geometric spreading assump-tion does.

We made finite-difference calculations of ∼1-Hz Pn

amplitudes for models with radial P-velocity gradients inthe mantle lid of 0:001 s�1, 0 s�1 (BEM), and �0:001 s�1.The rms amplitudes are shown in Figure 3. The BEM case

Figure 2. Pn amplitude versus distance curves for the BEM (redcurves, for no physical gradient in the lid) and for two models withpositive gradients in the mantle lid. The suite of curves for each casecorresponds to frequencies ranging from 0.75 Hz (lowest curve) to12 Hz (highest curve). The calculations were done with the reflec-tivity method, following the procedures of Yang et al. (2007).

Figure 3. Pn amplitude versus distance for the BEM (dia-monds), a 0:001 s�1 positive mantle gradient (triangles), and a�0:001 s�1 negative mantle gradient (squares).


(diamonds) corresponds well with the 1-Hz BEM amplitudesin Figure 2, and the positive gradient case (triangles) givesthe same shift of the amplitude curve as seen for the corre-sponding model 1-Hz reflectivity synthetics in Figure 2.The �0:001 s�1 case almost cancels out the EFT effect(0:0013 s�1), so this approaches the critical negative gradientthat would yield a flat-Earth model with constant half-spacevelocity and true head-wave behavior of Pn (see discussionin Yang et al., 2007). As expected, the amplitude decayapproaches a linear trend in log-amplitude versus log-distance for that situation.

Figure 2 indicates that it may be possible to use the Pn

geometric spreading representation of Yang et al. (2007) forthe BEM model and determine appropriate coefficients forany specific mantle lid velocity gradient that is larger thanthe critical negative gradient (�0:0013 s�1), but this is onlywarranted if there are sufficiently tight a priori constraints onthe specific model in the region for which the spreading is tobe utilized. As this would need to be done on a case-by-casebasis, we do not attempt to give generalized spreadingcorrections for varying mantle lid gradients here; the mainresult is recognizing the complex nature of the frequency-dependent shift that can result from the presence of smallmantle lid velocity gradients comparable to those observedin various regions.

Pn Sensitivity to Mantle P-Velocity Heterogeneity

Given the acute sensitivity of Pn geometric spreading toa usually poorly constrained attribute of the reference veloc-ity model (the radial velocity gradient in the uppermost man-tle lid), the stability of the complex behavior of Pn geometricspreading for the BEM in the presence of modest lateralheterogeneity is called into question. Most standard seis-mological practices assume rather simple reference models,known to be gross approximations of the real structure, underthe assumption that errors in the reference model produceonly mild biases in other estimated properties such asattenuation. This is simply not the case here, given the non-linear sensitivity of the Pn whispering gallery arrival to smallchanges in 1D structures. Clearly, Earth’s actual mantle lidhas multiscale lateral heterogeneities as well. Does theirpresence modify the acute sensitivity of Pn spreading relativeto the reference BEM structure?

We explore the effects of 2D lateral heterogeneity in themantle lid velocity structure by adding random lateral veloc-ity heterogeneity characterized by rms velocity fluctuationsof a particular strength distribution with random length-scalefluctuations exponentially distributed about prescribed hor-izontal (Ax) and vertical (Ay) length scales. This allows us toconsider 2D random heterogeneity models like that shownin Figure 4a, where the heterogeneities are superimposed

Figure 4. (a) Example realization of a velocity model with random P-wave velocity (VP) heterogeneity in the mantle lid. This model has0.5% rms random volumetric heterogeneities, exponentially distributed about a horizontal length scale of 40 km and a vertical length scale of3 km. (b) Example realization of a velocity model with random Moho depth fluctuations. This model has 3% rms random Moho depthvariations exponentially distributed about a horizontal length scale of 40 km.


on the Earth-flattened BEM. Because our finite-differencesimulations approximate a purely elastic system, randomvolumetric heterogeneities include rms perturbations ofP-wave velocity (VP), shear-wave velocity (VS), and density.For each set of random model parameters, VS varies by thesame percent rms as VP, while density has half the percentrms fluctuation as VS and VP. Analysis of the effects ofrandom heterogeneity requires a statistical sampling of theeffects associated with different realizations of the randomvelocity parameters. For each set of random parameters(Ax, Ay, % rms variability), we generate five velocity reali-zations, and calculate synthetic seismograms for individualrealizations. The ensemble average from all five simulationsis calculated to represent the result for a given set of randomparameters. This modest number of intensive calculationsappears to be sufficient to avoid significant bias in behaviorassociated with any single random configuration (Fig. 5). As

is always possible with random model parameters, thepossibility of outlier behavior and bias of the ensemble stillexists, and we do observe significant fluctuation in model Pn

amplitudes, so our focus will be on gross attributes of thecalculated Pn behavior, not on individual measurements.

Figure 5 shows the synthetic Pn amplitudes (∼1-Hz rmsamplitude in each 9-s-long Pn window) for five realizationsof structures with statistical properties corresponding tothose in Figure 4a, along with averages of the ensemble,and the finite-difference result for the BEM of Yang et al.(2007). An interesting behavior in Figure 5 is that the pres-ence of volumetric velocity fluctuations with large horizontalaspect ratios systematically affects the basic shape of the geo-metric spreading, reducing the minimum and allowing betterrepresentation by a linear log-log (power-law) type behaviorat this frequency. This suggests that the 1D geometric spread-ing behavior is rather delicate, with modest heterogeneitydisrupting the specific interference that gives rise to the com-plex shape of the amplitude curve with distance. It would beprohibitively expensive (and numerically challenging) toexplore the full frequency dependence of this behavior withcurrent computers, but it is likely that at higher frequenciesthe delicate energy partitioning of the direct Pn and thecoupled whispering gallery (as discussed by Yang et al.,2007) would be at least comparably disrupted.

Figure 6 compares the effect on Pn amplitude of variablepercent rms VP variation in the mantle lid for each of fourrandom heterogeneity configurations with aspect ratios ran-ging from isotropic (Fig. 6a: Ax � 10 km, Ay � 10 km)to large aspect ratio lenslike (Fig. 6d: Ax � 40 km,Ay � 3 km). As the aspect ratio increases, the deviationsfrom the BEM solutions tend to increase for weaker hetero-geneities. The overall amplitude of Pn also increases withincreased rms strength of VP variation. This is not surpris-ing—heterogeneities with stronger velocity contrasts areexpected to disrupt the wave field more effectively than moresubtle velocity fluctuations, expanding the portion of thewavefront that can be scattered into the Pn arrival window.This is illustrated in Figure 7 by seismic profiles of syntheticPn waveforms for the BEM, and for Earth models with largeaspect ratio Vp heterogeneity scale parameters correspond-ing to those in Figure 6d, with 1% and 2% rms VP fluctua-tions. Note the distortion of the Pn waveform and thepresence of significant early coda, indicating a high level offorward scattering of Pn energy within the mantle lid. Whilethe initial motion amplitudes are reduced relative to thehomogeneous model, the overall rms values of the full win-dow increase, indicating that signals with a wider range ofray parameters are being scattered into the window.

It is also interesting to note the difference in the shape ofthe amplitude curves in Figure 6 as the strength of hetero-geneity increases. The same measurements are regroupedin Figure 8 by the windowing method, where the averaginglengths vary for models with common levels of 0.5%(Fig. 8a,d,g), 1% (Fig. 8b,e,h), and 2% rms (Fig. 8c,f,i) VP

fluctuation. In the 0.5% case, the curve is concave-up for the

Pn

Figure 5. Pn amplitude at ∼1 Hz plotted with distance for fiverealizations of random mantle lid volumetric VP heterogeneity withthe same averaging length scales and rms strength distributionshown in Figure 4a (Ax � 40 km, Ay � 3 km, RMS � 0:5%).The averages (circles) of these five random realizations are shown,as well as the Pn amplitudes for the BEM (diamonds).


more isotropic heterogeneities and approximately linear forthe more lenslike heterogeneities. When the rms VP fluctua-tion is 1%, there is a stronger increase in Pn amplitude as the

averaging lengths increase in aspect ratio, with the appear-ance of a peak around 900 km distance. In the 2% rms VP

variation cases, there is less variation with aspect ratio, and

Pn

Pn

Figure 6. Ensemble-averaged Pn amplitudes as a function of distance for all configurations of random volumetric mantle heterogeneitysimulated. Pn amplitude for the BEM (diamonds) are compared to models with 0.5% rms (squares), 1.0% rms (triangles), and 2% rms (×s)VP fluctuation for horizontal (Ax) and vertical (Ay) averaging lengths of (a) Ax � 10 km, Ay � 10 km (isotropic); (b) Ax � 20 km,Ay � 10 km; (c) Ax � 20 km, Ay � 6 km; and (d) Ax � 40 km, Ay � 3 km. Each value is the average of five realizations with differentseed kernels.


the amplitudes at distances from approximately 400–900 kmincrease, effectively smoothing out the local amplitude peakat 900 km, as is apparent in the 1% cases. The calculationsfor 2% rms VP heterogeneity approach a linear (power-law)trend in log-amplitude versus log-distance, with the slopebecoming more steeply negative with increasing aspect ratioof the mantle lid heterogeneity. It might be tempting to viewthese results as confirmation of the validity of the power-lawgeometric spreading approximation, but these modelingresults correspond to a specific Earth model, as would anylinear power-law fit to them. Furthermore, if a log-linear fit toany specific model case was specified, it would not work formost other cases—it would be specific to a single Earthstructure or a narrow suite of structural configurations, thefine-scale of which is unconstrained in the actual Earth.

As Figure 6 and Figure 8 clearly demonstrate, Pn

amplitudes in most model cases increase in response to a het-erogeneous mantle lid. The degree to which the amplitudesincrease and approach a log-linear trend varies from case tocase and by windowing method, as does the rate of amplitudedecay. If Pn scattering attenuation were to be included in anapparent attenuation model instead of being inadvertentlyremoved by the geometric spreading approximation, it mightbe possible in some regions of strong scattering to producemodels with negative apparent attenuation. However, Yanget al. (2007) applied their BEM geometric spreading correc-tion to a large data set for Eurasia and obtained very reason-able (nonnegative) values, so it seems likely that this wouldonly be the case for very extreme cases.

Each of the amplitude values shown in Figure 6 andFigure 8 are the average of five individual simulations withrandom configurations for a given set of parameters, thus theoverall behavior is not the effect of any specific structural

configuration. For example, the peak in Pn amplitude near900 km for VP fluctuations of 1% (Fig. 8b,e,h) is likely theresult of a complex relationship between the strength of theVP heterogeneities and the angle at which Pn energy pene-trates into the uppermost mantle. Mantle lid VP heterogene-ities of ∼1% are favorable for concentrating horizontallyrefracted energy, particularly when higher aspect ratio het-erogeneities are present. Stronger heterogeneities do thiseffectively with little dependence on the shapes of the het-erogeneities. Energy that penetrates into the lid with higherangles of incidence is unlikely to be trapped by these randomheterogeneities unless very strong features are immediatelyencountered. The heterogeneous structures thus capture alimited range of ray parameters, and the increase in ampli-tude at 900 km represents a tuning of the refractions capturedfor a specific level of heterogeneity for 1-Hz waves. Thisinteraction between angle of incidence and strength of veloc-ity fluctuation likely underlies why only the strongest VP

heterogeneities have strong effect on Pn amplitudes at allranges (Fig. 8c,f,i) and why the higher aspect ratio hetero-geneities (Fig. 6c,d) have the highest Pn amplitude levelsfor all distances. The case of Pn amplitude calculated inthe short fixed time window (Fig. 8g,h,i) is an exceptionto this pattern, as Pn has the lowest amplitudes at distancesgreater than 1000 km for 2% VP fluctuations of all aspectratios, which become increasingly lowered with heterogene-ities of increasing aspect ratio. This is the result of the shortwindow excluding all of the Pn coda following the initial Pn

arrival, and reflects the shift in energy distribution resultingfrom increased mantle heterogeneity. The longer timewindows (group velocity and fixed window) show increasedamplitude with higher aspect ratio because they are domi-nated at greater distances by the scattered Pn energy that

(a) (b) (c)

Figure 7. Synthetic Pn seismograms generated in (a) the BEM; (b) a velocity model with 1% rms random VP fluctuations with horizontalaveraging length scale of 40 km and a vertical averaging length scale of 3 km; and (c) a velocity model with 2% rms random VP fluctuationson the same averaging length scales as (b). Each trace is normalized on the largest amplitude.


is captured and returned to the surface by the heterogeneousvelocity structure, whereas the short window shows greatlydecreased amplitude of the initial Pn arrival at greater dis-tances. Regardless of the windowing scheme used, the stron-gest heterogeneities with the highest aspect ratios have the

most dramatic effects on Pn amplitude. The specific geo-metric spreading behavior of the BEM or the 1D gradientmodels is clearly a result of rather delicate energy partition-ing between diving energy and the suite of multiply reflectedunderside Moho reflections that is corrupted by the presence

Homogeneous EarthAx=10 km, Ay=10 kmAx=20 km, Ay=10 kmAx=20 km, Ay=6 kmAx=40 km, Ay=3 km

RMS = 2.0%, All A’sRMS = 1.0%, All A’sRMS = 0.5%, All A’s

3 sec. Short Window

8.2-7.6 km/sec Group Velocity

9 sec. Fixed Window

Distance (km)

1000200 2000

1.0

0.1

0.01

Distance (km)

1000200 2000

1.0

0.1

0.01

Pn

am

plit

ud

e (x

10-5

)

Distance (km)

1000200 2000

1.0

0.1

0.01

1000200 2000

1.0

0.1

0.011000200 2000

1.0

0.1

0.011000200 2000

1.0

0.1

0.01

1000200 2000

1.0

0.1

0.011000200 2000

1.0

0.1

0.011000200 2000

1.0

0.1

0.01

Pn

am

plit

ud

e (x

10-5

)P

n a

mp

litu

de

(x10

-5)

(b)(a) (c)

(e)(d) (f)

(h)(g) (i)

Figure 8. Ensemble-averaged Pn amplitudes as a function of distance for all configurations of random volumetric mantle heterogeneity(Fig. 6), grouped by percent rms fluctuation and windowing method. (a–c) Amplitudes calculated with a 9-s fixed time window; (d–f) am-plitudes calculated with a group velocity window (8:2–7:6 km=s); and (g–i) amplitudes calculated with a short (3-s) fixed window positionedbased the onset of Pn. Pn amplitude for the BEM is shown (diamonds) for comparison to heterogeneity with horizontal and vertical averaginglengths of: 10 × 10 km (squares), 20 × 10 km (triangles), 20 × 6 km (×s), and 40 × 3 km (asterisks) for (a, d, g) 0.5% rms VP fluctuations;(b, e, h) 1% rms VP fluctuations; and (c, f, i) 2% rms VP fluctuations.


of small velocity fluctuations and associated multipathing.We now examine whether a rough Moho boundary has com-parable effects.

Pn Sensitivity to Moho Topography

Volumetric heterogeneity in the mantle lid is expected toresult from geological processes during crustal formation andevolution, and the same is true for Moho boundary irregu-larities on multiple scales. Pn interactions with the Mohonear the source and the receiver, as well as throughout themantle lid propagation and whispering gallery development,should yield Pn sensitivity to the Moho roughness. Modelswith large-scale variations in crustal thickness are available,and one can make geometric spreading calculations for a spe-cific deterministic model if the structure is known; however,relatively little constraint exists on the spectrum of small-scale Moho irregularities likely to affect any specific Pn

phase or average behavior within a given region. We haveparameterized 2D exponential-distribution statistical irregu-larities in the Moho depth (a realization of which is shownin Fig. 4b) in otherwise homogeneous (BEM) velocity layer-ing to explore the effects of random Moho topographicirregularities on Pn geometric spreading. We conductedsimulations with Gaussian-distributed random Moho irregu-larity as well, finding comparable overall simulation resultsfor a 1-s dominant period. The main difference between thedistributions is that the exponential model is much richer insmall-scale structure and produces longer persisting codathan the smoother Gaussian model.

We created velocity models with Moho irregularityhaving four different averaging length scales, with randomhorizontal length scale exponentially distributed about dis-tances ranging from 20 to 160 km. Topography of the Mohoboundary is modeled with 1% (�0:4 km), 3% (�1:2 km), or5% (�2:0 km) rms fluctuations for each of the distancescales. There are few observational constraints on the appro-priate statistics for Moho irregularities in general, but thesevalues are representative of fluctuations in reflection seis-mology imaging. The average Moho depth always remains40 km, as in the BEM. Results of these simulations are shownin Figure 9, grouped by horizontal averaging length, and inFigure 10, grouped by rms height of Moho topographyand windowing scheme. These are again ensemble averagevalues from five different model realizations for each set ofrandom parameters. In Figure 9a, for 20-km averaginglength, the 3% height variation actually reduces the averagePn amplitudes to below those for the BEM. The strongestpositive perturbations to Pn amplitudes relative to the BEMresult from Moho variations distributed around the 40-kmaveraging length scale (Fig. 9b). For this horizontal aver-aging scale, the amplitude scatter becomes significant for5% rms heterogeneity in the Moho topography. There is onlyminor sensitivity to the choice of horizontal averaging func-tion for values larger than 40 km.

Figure 10 displays the effects of percent topography fluc-tuation more clearly, with 1% rms variability having littleeffect on Pn amplitude for any of the horizontal length scalessimulated and 5% height fluctuation clearly disrupting the Pn

geometric spreading pattern relative to the BEM. The disrup-tion mainly involves increased amplitude scatter without asystematic shift in the pattern, as was found in the volumetricheterogeneity simulations. Moho topographic variability of3%–5% is required for all horizontal averaging scales inorder to produce significant scatter in Pn amplitudes. Weattribute most of this amplitude variability to the sensitivityof the ray paths for Pn as it first encounters and refracts atthe Moho, although progressive scattering along the horizon-tal path contributes overall. The consistency of the Pn ampli-tude calculations (relative to the homogenous model case)across all windowing methods confirms that fluctuations inamplitude are mostly controlled by the first refraction of Pn

at the Moho, and the effect of forward-scattered wave energyalong the Moho on Pn amplitudes is relatively minor, ascompared to the amplitudes calculated in the model casesof mantle heterogeneity, which are dominated by scatteredPn

energy. The amount of forward-scattered Pn energy dependson the direction that the boundary is dipping below the source,as well as how steeply it is dipping, and even with five realiza-tions being averaged, there is rapid fluctuation in amplitudesthat emerge at various distances. When the synthetic wave-forms are examined (Fig. 11), one can see that Moho varia-bility strongly affects Pn arrival time (mainly because thecrustal thickness is locally thicker or thinner near the sourceand receiver, depending on the specificMoho topography) butalso results in moderate scattering of Pn energy, as evidencedby the enhancedPn coda following the first arrival. Given thatall of the scattering in these models must originate at theMoho, the coda levels are influenced by both near-sourceand near-receiver scattering along with scattering of energyout of the whispering gallery of underside Moho reflections.

Discussion and Conclusions

The behavior of Pn geometric spreading is subtle andcomplex, even for simple 1D structures (e.g., Menke andRichards, 1980; Sereno and Given, 1990; Yang et al., 2007).The calculations in this study provide a basic explanation forthe high variance observed in Pn rms amplitudes fromthe ANOVA analysis (Fisk et al., 2008) mentioned in theIntroduction. Variable 1D and 2D structures readily producenonlinear, frequency-dependent variations superimposed onthe already complex behavior for the simple BEM. This var-iability is somewhat analogous to the higher variance foundformb measurements for teleseismic Pwaves versus the less-variable mb�Lg� for complex, but heavily path-averagingregional Lg phases; the energy in the Pn window is com-prised of a fairly narrow range of ray parameter arrivals thatare very sensitive to the structure along any specific path.

This study demonstrates that the effects of variable 1Dmantle lid P velocity gradients preserve the basic functional


form of the frequency-dependent complexity of the constant-velocity mantle lid model, with positive gradients causing ashift of amplitude curves toward higher frequencies for thezero-gradient BEM. The effects are so strong that specific

frequency-dependent coefficients would need to be deter-mined for the spreading model for any specific gradient.The nonlinear sensitivity raises questions about the choiceof spreading model given typical situations where there is

(a) (b)

(c) (d)

Pn

Pn

Figure 9. Ensemble-averaged Pn amplitudes plotted as a function of distance for all configurations of random Moho topography hetero-geneity, grouped by horizontal averaging length scale, Ax. Pn amplitude for the BEM (diamonds) are compared to amplitudes for simulationswith 1% rms (squares), 3% rms (triangles), and 5% rms (×s) Moho depth variations for horizontal averaging length scales of (a) 20 km,(b) 40 km, (c) 80 km, and (d) 160 km.


little or no information on the local mantle lid velocitygradients. Of course, if the data for a given region are suffi-cient to constrain the mantle lid velocity gradient (as may bethe case across the Russian platform; e.g., Morozova et al.,1999), specific spreading computations can provide a localreference behavior relative to which attenuation estimates

can more reliably be determined. In practice, this is still chal-lenging, given the degenerate nature of layered versus gra-dient model fits to first-arrival times.

Computations with lateral volumetric heterogeneities inthe mantle lid and statistical irregularities in Moho topogra-phy add significant complexity to the general problem of

RMS = 5.0%, All A’sRMS = 3.0%, All A’sRMS = 1.0%, All A’s

3 sec. Short Window

8.2-7.6 km/sec Group Velocity

9 sec. Fixed Window

Distance (km)

1000200 2000

1.0

0.1

0.01

Distance (km)

1000200 2000

1.0

0.1

0.01

Pn

am

plit

ud

e (x

10-5

)

Distance (km)

1000200 2000

1.0

0.1

0.01

1000200 2000

1.0

0.1

0.011000200 2000

1.0

0.1

0.011000200 2000

1.0

0.1

0.01

1000200 2000

1.0

0.1

0.011000200 2000

1.0

0.1

0.011000200 2000

1.0

0.1

0.01

Pn

am

plit

ud

e (x

10-5

)P

n a

mp

litu

de

(x10

-5)

Homogeneous EarthAx=20 kmAx=40 kmAx=80 kmAx=160 km

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 10. Ensemble-averaged Pn amplitudes plotted as a function of distance for all configurations of random Moho topographyheterogeneity, grouped by percent rms depth fluctuation and windowing method. (a–c) Amplitudes calculated with a 9-s fixed time window;(d–f) amplitudes calculated with a group velocity window (8:2–7:6 km=s); and (g–i) amplitudes calculated with a short (3-sec) fixed windowpositioned based the onset of Pn. Pn amplitude for the BEM (diamonds) are compared to horizontal averaging length scales of 20 km(squares), 40 km (triangles), 80 km (×s), and 160 km (asterisks), for percent rms depth fluctuations of 1% for (a, d, g); 3% for (b, e,h); and 5% for (c, f, i).


approximating Pn geometric spreading. Our 2D finite-difference simulations indicate that the precise shape of thegeometric spreading behavior for 1D models can be disruptedby even moderate levels of lateral heterogeneity, and thebehavior for 1-s period waves moves toward a power-lawrepresentation as the level of volumetric heterogeneityincreases and as the horizontal-to-vertical aspect ratio of theheterogeneities increases. This behavior may provide a ratio-nale for using a conventional power-law type geometricspreading correction, but how closely such a model approx-imates actual Pn behavior depends critically on the level andtype of heterogeneity present in the real Earth. For example,simulations with subtle mantle heterogeneity (0.5% rms VP

fluctuations; Fig. 8a,d,g) indicate that Pn geometric spread-ing can be more closely approximated by the 1D homoge-neous BEM relations determined by Yang et al. (2007),while simulations with stronger mantle heterogeneity(Fig. 8c,f,i) can be better approximated by a power-law typecorrection, at least for 1 Hz, although a single linear decayrate would not fit all cases approaching a log-linear decay.The Pn amplitudes of simulations through velocity modelsbetween these two end members cannot be particularly wellfit by either type of geometric spreading correction.

For rough Moho topography heterogeneity, departuresfrom the 1D Pn geometric spreading pattern prove muchless significant. Models with 3%–5% topographic fluctua-tions over scale lengths of approximately 80 km seem tohave the greatest effect on Pn amplitude near 1 Hz, but,for nearly all simulations, the BEM geometric spreadingcorrection of Yang et al. (2007) would be an acceptable re-ference model.

These results present something of a conundrum: the 1DBEM structure considered by Yang et al. (2007) is very sim-ple but produces complex, frequency-dependent, geometricspreading behavior. Our results show that the behavior isnonlinearly sensitive to the model parameters, and the 1Dgradient models indicate that this extends to the frequencydependence. However, our 2D models with the strongest,but still geologically plausible, mantle heterogeneities pro-duce relatively simple patterns of geometric spreading, atleast for 1 Hz. It is likely that Pn simulations over a broaderfrequency band and for 3D structures would also have highsensitivities to different scale lengths and velocity contrastsas a function of frequency. The effects of 2D or 3D VP

heterogeneity on shorter period signals have not been quan-titatively assessed for the long propagation distances consid-ered here; doing so presents formidable computationalchallenges. We speculate that eventually simulations inheterogeneous 3D structures will enhance the tendenciesmanifested in our 2D calculations.

Prior studies of Pn attenuation efforts have usually as-sumed simple frequency-independent power-law geometricspreading behavior for Pn (e.g., Sereno et al., 1988; Pasya-nos et al., 2009), which is not a realistic form of spreading forany known 1D (i.e., laterally homogeneous) velocity model.This assumption may actually yield reasonable results to theextent that the specific power-law corresponds to a fairlyhigh degree of uppermost mantle velocity heterogeneity inthe real 3D Earth, which may be evidenced in the data bythe amount of coda following the initial Pn arrival (Fig. 7and Fig. 11). Of course, the precise spectrum and statisticaldistribution of real mantle lid heterogeneities is not known

(a) (b) (c)

Figure 11. Finite-difference synthetic Pn seismograms generated in (a) BEM; (b) a velocity model containing 3% rms random Mohodepth variation with horizontal averaging scale length of 40 km; and (c) a velocity model with 5% rms random Moho depth variation alonghorizontal averaging scale length of 40 km. Each trace is normalized on the largest amplitude.


for any specific path, just as the precise best 1D lid velocitygradient is not reliably known, so it would be difficult toestimate a particular small-scale structural configuration witha power-law geometric spreading approximation that has anappropriate decay rate. That situation is unlikely to change ingeneral. Conventionally, seismic reference models used forcomputing geometric spreading of seismic signals with alimited ray parameter range are selected to be very simplestructures rather than such complex (and unspecified) struc-tures that the wave field is scattered to the point of allowingan energy flux power-law assumption to be made (as is quitereasonably done for the multiray parameter phases like Lg).It should be recognized that if one assumes (arbitrarily) apower-law form of geometric spreading for Pn, the underly-ing reference Earth model is explicitly not a conventional 1Dconstant-velocity mantle model or even a constant lid-gradient model, but is intrinsically a statistically heteroge-neous model with unspecified properties. It is ratheruncomfortable to invoke such a model. In addition, werecognize that even though scattering due to velocity hetero-geneities is an elastic effect, the single-wavefront definitionof geometric spreading is often assumed, and elastic scatter-ing effects are considered part of the apparent attenuation.

Our recommendation is that, in the absence of specificregional constraints on the mantle lid velocity gradient andon small-scale heterogeneity in the lid and in Moho topogra-phy, the best choice for Pn geometric spreading is thefrequency-dependent version found for the BEM by Yanget al. (2007). This has the primary merit of having a known,simple 1D Earth model associated with the geometric spread-ing behavior and a parameterization that can accommodatethe nonlinear response of Pn to large-scale changes in theEarth velocity model (e.g., mantle gradient), which is con-sistent with standard seismological modeling practices. Ifthis model is used in determining an apparent attenuationstructure, the attenuation model will very likely have explicitfrequency dependence. As is always the case, deficiencies inthe spreading corrections will likely project into the attenua-tion model, but, as long as the resulting attenuation model isviewed as a collective parameterization of intrinsic P waveattenuation, correction for scattering attenuation (whichcould involve negative Qs), and frequency-dependent errorsin the geometric spreading relations (which, based on ourmodeling results, could approach a factor of 10 for any rela-tion chosen), at least the attenuation model is defined relativeto a known elastic Earth model. With most applicationsinvolving a convolution of the geometric spreading andattenuation models, the trade-offs will not bias sourcestrength estimates; but one must exercise care in specificinterpretation of attenuation parameters in terms of Earthproperties. Decoupling geometric spreading and attenuationcan only occur with a priori constraint on the structure (e.g.,using travel-time curves to constrain the upper-mantle veloc-ity gradient), but small-scale structure of the type modeledhere is largely unconstrained. Investigators should be awarethat by adopting a specific geometric spreading approxima-

tion, they are also adopting assumptions about Earth struc-ture. It is therefore imperative that investigators documentthe assumptions inherent in any geometric spreading approx-imation chosen in attenuation modeling studies to preventassumptions specific to a given study region from being er-roneously applied to other regions. It could be argued that,with amplitude variations between a heterogeneous Earthmodel and a homogeneous Earth model as large as they are,an approximation of Pn geometric spreading should in someway account for scattering due to small-scale heterogeneity.This is problematic, as the small-scale heterogeneity is notknown. The data can serve as somewhat of a guide, withobserved Pn-Pn coda variations indicating a strong or weakscattering domain, but quantitatively linking the data to aspecific heterogeneous model remains beyond our scope asa community. By continuing efforts to constrain the mantlelid velocity gradient empirically, or to use independentknowledge of the Moho topography, calculations of geo-metric spreading for corresponding models can be performedto enable better resolution of true attenuation effects.

It is clear from these simulations that either the 1D BEMgeometric spreading corrections of Yang et al. (2007) or anypower-law correction will have significant uncertainty inapproximating the actual geometric spreading of Pn for spe-cific data sets; neither is ideal for all cases. The extent of thisuncertainty is difficult to quantify, given the lack of con-straints on the small-scale seismic velocity heterogeneityspectrum within the real Earth. Thus, propagating realisticuncertainties in the geometric spreading model into formaluncertainties in attenuation model estimates remains verydifficult. We seek to perform future calculations for 3D struc-tures with greater bandwidth, with the goal of providinguseful quantification of the uncertainties in attenuation mod-eling due to geometric spreading approximation.

Data and Resources

All data used in this paper came from published sourceslisted in the references.

Acknowledgments

We thank George Randall of Los Alamos National Laboratory forassisting us with his reflectivity code. We thank Yaofeng He, for helpfuldiscussion and advice on the two-dimensional finite-difference calculations.Many thanks also to Bill Walter and Igor Morozov for their insightful re-views. This work was performed under the auspices of the U.S. Departmentof Energy by Los Alamos National Laboratory and University of CaliforniaSanta Cruz under contracts DE-AC52-06NA25396 and DE-FC52-05NA26606.

References

Buldyrev, V. S., and A. I. Lanin (1965). Interfering waves at the surface of anelastic inhomogeneous sphere, Rev. Geophys. 3, 49–54.

Chapman, C. H. (1973). The Earth flattening transformation in body wavetheory, Geophys. J. R. Astr. Soc. 35, 55–70.


Fisk, M. D., S. R. Taylor, T. Lay, and X. Xie (2008). Analysis and modelingof regional seismic phases at nuclear test sites: Implications for P=Sdiscrimination, NNSA Report MRC/WDC-R-560, 99 pp.

Hartse, H. E., S. R. Taylor, W. S. Phillips, and G. E. Randall (1997). A pre-liminary study of regional seismic discrimination in central Asia withemphasis on western China, Bull. Seismol. Soc. Am. 87, 551–568.

Hill, D. P. (1973). Critically refracted waves in a spherically symmetricradially heterogeneous Earth model, Geophys. J. R. Astr. Soc. 34,149–177.

Menke, W. H., and P. G. Richards (1980). Crust-mantle whispering galleryphases: A deterministic model of teleseismic Pn wave propagation,J. Geophys. Res. 85, 5416–5422.

Morozov, I. (2010). On the causes of frequency-dependent apparent seismo-logicalQ, Pure Appl. Geophys., 167, no. 10, 1131–1146, doi 10.1007/s00024-010-0100-6.

Morozova, E. A., I. B. Morozov, and S. B. Smithson (1999). Heterogeneityof the uppermost mantle beneath Russian Eurasia from the ultra-long-range profile QUARTZ, J. Geophys. Res. 104, 20,329–20,348.

Müller, G. (1977). Earth-flattening approximation for body waves derivedfrom geometric ray theory—Improvements, corrections and range ofapplicability, J. Geophys. 42, 429–436.

Myers, S. C., M. L. Begnaud, S. Ballard, M. E. Pasyanos, W. S. Phillips,A. L. Ramirez, M. S. Antolik, K. D. Hutchenson, J. J. Dwyer,C. A. Rowe, and G. S. Wagner (2010). A crust and upper-mantle modelof Eurasia ad North Africa for Pn travel-time calculation, Bull.Seismol. Soc. Am. 100, 640–656, doi 10.1785/0120090198.

Pasyanos, M. E., W. R. Walter, and E. M. Matzel (2009). A simultaneousmultiphase approach to determine P-wave and S-wave attenuation ofthe crust and upper mantle, Bull. Seismol. Soc. Am. 99, 3314–3325,doi 10.1785/0120090061.

Randall, G. E. (1994). Efficient calculations of complete differential seis-mograms for laterally homogeneous earth models, Geophys. J. Int.118, 245–254.

Sereno, T. J., and J. W. Given (1990). Pn attenuation for a sphericallysymmetric Earth model, Geophys. Res. Lett. 17, 1141–1144.

Sereno, T. J., S. R. Bratt, and T. C. Bache (1988). Simultaneous inversion ofregional wave spectra for attenuation and seismic moment in Scandi-navia, J. Geophys. Res. 93, 2019–2036.

Taylor, S. R., A. A. Velasco, H. E. Hartse, W. S. Phillips, W. R. Walter, andA. J. Rodgers (2002). Amplitude corrections for regional seismic dis-criminants, Pure Appl. Geophys. 159, 623–650.

Walter, W. R., and S. R. Taylor (2001). A revised magnitude and distanceamplitude correction (MDAC2) procedure for regional seismic discri-minants: Theory and testing at NTS, Lawrence Livermore NationalLaboratory Report UCRL-ID-146882, 13 pp.

Walter, W. R., K. M. Mayeda, and H. Patton (1995). Phase andspectral ratio discrimination between NTS earthquakes andexplosions. Part I: Empirical observations, Bull. Seismol. Soc. Am.85, 1050–1067.

Xie, X., and T. Lay (1994). The excitation of Lg waves by explosions: Afinite-difference investigation, Bull. Seismol. Soc. Am. 84, 324–342.

Yang, X. (2002). A numerical investigation of Lg geometric spreading, Bull.Seismol. Soc. Am. 92, 3067–3079.

Yang, X., T. Lay, X. Xie, and M. S. Thorne (2007). Geometric spreading ofPn and Sn in a spherical Earth model, Bull. Seismol. Soc. Am. 97,2053–2065.

Zhao, L. S., and J. Xie (1993). Lateral variations in compressional velocitiesbeneath the Tibetan Plateau from Pn travel time tomography,Geophys.J. Int. 115, 1070–1084.

Department of Earth and Planetary SciencesUniversity of California, Santa Cruz1156 High StreetSanta Cruz, California 95064

(M.A., T.L.)

Institute of Geophysics and Planetary PhysicsUniversity of California, Santa CruzSanta Cruz, California 95064

(X.-B.X.)

Geophysics GroupEarth and Environment Sciences DivisionLos Alamos National LaboratoryLos Alamos, New Mexico 87545

(X.Y.)

Manuscript received 27 April 2010


http://dx.doi.org/10.1007/s00024-010-0100-6

http://dx.doi.org/10.1007/s00024-010-0100-6

http://dx.doi.org/10.1785/0120090198

http://dx.doi.org/10.1785/0120090061

and Moho Topography on P Geometric Spreadingthorne/TL.pdfs/ALXY_Pn_BSSA_2011.pdf · Effects of 2D Random Velocity Heterogeneities in the Mantle Lid and Moho Topography on P n Geometric

Documents